is known as cut locus of the point \(p \in N\)  and each point in this set is called a cut point of \(p\) . Moreover, the cut locus of \(p\)  coincides with the closure of the set of those points \(q \in N\)  such that there is more than one distance minimizing geodesics from \(p\)  to \(q\)  (see for instance [1]}). It has been also shown in [2]} that \(d(p,\cdot )\)  is smooth in \(N\setminus (\lbrace p\rbrace \cup {\text{cut}}_N(p))\)  but not at any \(q \in (\lbrace p\rbrace \cup {\text{cut}}_N(p))\) . In order to understand the smoothness properties of the distance function on a
Riemannian manifold \((M,g)\)  with a strictly convex boundary, our aim is to define the set analogous to the one in (REF ) in this context.
